Mathematics and Computing for Scientific Innovation Vol. 1

The transport of carbon dioxide (CO₂) through reactive porous media involves the simultaneous interplay of molecular diffusion, bulk advection, and concentration-dependent chemical reactions. Modelling this system accurately requires numerical schemes that are not only computationally efficient but are also rigorously verified to be stable and consistent. In this chapter, the numerical properties of an explicit DuFort–Frankel finite difference scheme applied to a dimensionless reaction–diffusion–advection (RDA) equation that incorporates a linear reactive source term, a linear sink term, and a uniform external source are analysed. Starting from the standard three-point central-difference stencil in space and the DuFort–Frankel time-averaged update in time, we derive the matrix representation of the discrete system. Stability is examined by two complementary routes. Analytically, the amplification factor satisfies a quadratic characteristic equation whose roots are shown to obey a modulus bound of unity for every admissible Fourier mode and every mesh ratio, so that the diffusive march is unconditionally stable; the centred advection term preserves this bound provided that the cell Péclet number does not exceed two. Numerically, the eigenvalues of the amplification matrix were computed in MATLAB for a physically representative parameter set drawn from the In Salah CO₂ storage project, and all eight eigenvalues were found to lie strictly inside the unit circle. Consistency is verified through a formal Taylor series expansion of the discrete stencil about the exact solution. The local truncation error is second-order in both space and time, but the DuFort–Frankel time averaging introduces an additional term involving the ratio of the time step to the grid spacing. The scheme is therefore conditionally consistent, recovering the parabolic limit only when the time step is refined faster than the grid spacing. Subject to that refinement path, the Lax equivalence theorem yields second-order convergence. These findings establish the discretisation as a mathematically sound tool for simulating CO₂ transport dynamics and explicitly state the conditions under which it may be relied upon.